Trig Calculator
Find sin, cos, tan, csc, sec and cot of any angle in degrees or radians, with exact values like √3/2, the reference angle, quadrant, and inverse trig.
Trig Calculator: with the default inputs, result is sin 30° = 1/2, cos 30° = √3/2, tan 30° = √3/3.
Any angle, positive or negative. For π/6 rad type 0.5235987756.
Used only for inverse trig.
The ratio you know, e.g. sin θ = 0.5.
- sin θ
- 0.5
- cos θ
- 0.866025
- tan θ
- 0.57735
- csc θ
- 2
- sec θ
- 1.154701
- cot θ
- 1.732051
- Exact values
- sin = 1/2, cos = √3/2, tan = √3/3, csc = 2, sec = 2√3/3, cot = √3
- Angle in degrees
- 30
- Angle in radians
- π/6 ≈ 0.523599
- Reference angle (degrees)
- 30
- Quadrant
- Quadrant I — all six positive
Assumptions
- Angles are in degrees or radians; gradians are not offered.
- Exact radical forms are given for multiples of 15°; other angles are shown to six decimal places.
- Inverse functions return principal values: sin⁻¹ and csc⁻¹ in [−90°, 90°], cos⁻¹ and sec⁻¹ in [0°, 180°], tan⁻¹ in (−90°, 90°), cot⁻¹ in (0°, 180°).
| Degrees | Radians | Point (cos, sin) | sin | cos | tan | |
|---|---|---|---|---|---|---|
| 0° | 0 | (1, 0) | 0 | 1 | 0 | |
| 30° | π/6 | (√3/2, 1/2) | 1/2 | √3/2 | √3/3 | ◀ your angle |
| 45° | π/4 | (√2/2, √2/2) | √2/2 | √2/2 | 1 | |
| 60° | π/3 | (1/2, √3/2) | √3/2 | 1/2 | √3 | |
| 90° | π/2 | (0, 1) | 1 | 0 | undefined | |
| 120° | 2π/3 | (−1/2, √3/2) | √3/2 | −1/2 | −√3 | |
| 135° | 3π/4 | (−√2/2, √2/2) | √2/2 | −√2/2 | −1 | |
| 150° | 5π/6 | (−√3/2, 1/2) | 1/2 | −√3/2 | −√3/3 | |
| 180° | π | (−1, 0) | 0 | −1 | 0 | |
| 210° | 7π/6 | (−√3/2, −1/2) | −1/2 | −√3/2 | √3/3 | |
| 225° | 5π/4 | (−√2/2, −√2/2) | −√2/2 | −√2/2 | 1 | |
| 240° | 4π/3 | (−1/2, −√3/2) | −√3/2 | −1/2 | √3 | |
| 270° | 3π/2 | (0, −1) | −1 | 0 | undefined | |
| 300° | 5π/3 | (1/2, −√3/2) | −√3/2 | 1/2 | −√3 | |
| 315° | 7π/4 | (√2/2, −√2/2) | −√2/2 | √2/2 | −1 | |
| 330° | 11π/6 | (√3/2, −1/2) | −1/2 | √3/2 | −√3/3 |
Each point on the unit circle is (cos θ, sin θ). Memorise the first quadrant (0°, 30°, 45°, 60°, 90°) and the rest follows from the reference angle and the quadrant's signs: All Students Take Calculus — all, sin, tan, cos positive in quadrants I to IV.
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How this is worked out
The formula
On the unit circle, the point at angle θ is (cos θ, sin θ) tan θ = sin θ / cos θ cot θ = cos θ / sin θ sec θ = 1 / cos θ csc θ = 1 / sin θ radians = degrees × π / 180 reference angle: θ (QI), 180° − θ (QII), θ − 180° (QIII), 360° − θ (QIV)
Open How it’s calculated above to see this worked through with your own numbers.
What you enter
- What do you want to find?
- Choose one of 2 options.All six trig functions of an angle · The angle from a trig value (inverse trig)
- Angle
- Any angle, positive or negative. For π/6 rad type 0.5235987756.in deg, rad · defaults to 30
- Inverse function
- Used only for inverse trig.sin⁻¹ (arcsin) · cos⁻¹ (arccos) · tan⁻¹ (arctan) · csc⁻¹ (arccsc) · sec⁻¹ (arcsec) · cot⁻¹ (arccot)
- Trig value
- The ratio you know, e.g. sin θ = 0.5.defaults to 0.5
What you get back
- Resultmain answer
- sin θ
- cos θ
- tan θ
- csc θ
- sec θ
- cot θ
- Exact values
- Angle in degrees
- Angle in radians
- Reference angle (degrees)
- Quadrant
- All solutions
- Inverse trig only: every angle with that value.
What this assumes
- Angles are in degrees or radians; gradians are not offered.
- Exact radical forms are given for multiples of 15°; other angles are shown to six decimal places.
- Inverse functions return principal values: sin⁻¹ and csc⁻¹ in [−90°, 90°], cos⁻¹ and sec⁻¹ in [0°, 180°], tan⁻¹ in (−90°, 90°), cot⁻¹ in (0°, 180°).
About this calculator
This trig calculator gives all six trigonometric functions of an angle — sine, cosine, tangent, cosecant, secant and cotangent — in degrees or radians, and runs the other way too: give it a value like sin θ = 0.5 and it returns the angle. For the special angles every course asks about it prints the exact answer (√3/2, not 0.866025), which is what homework and exams want.
How to use it
Choose "All six trig functions of an angle", type the angle and pick degrees or radians from the unit menu. Negative angles and angles past 360° are fine; the calculator reduces them to the coterminal angle between 0° and 360° and shows that step. For inverse trig, switch the mode, choose sin⁻¹, cos⁻¹, tan⁻¹ or one of the reciprocal inverses, and enter the known ratio.
Reading the results
- The six values to six decimal places. Where a function divides by zero — tan 90°, sec 270°, cot 0°, csc 180° — it says undefined rather than printing a huge number, because the function genuinely has no value there.
- Exact values appear whenever the angle is a multiple of 15°. Multiples of 30° and 45° give the familiar ½, √2/2 and √3/2; 15° and 75° give the less familiar (√6 − √2)/4 forms from the half-angle identities.
- Reference angle and quadrant. The reference angle is the acute angle to the x-axis. Every trig value equals the value at the reference angle, with a sign set by the quadrant — which is why knowing 30°, 45° and 60° is enough to do the whole circle.
- All solutions (inverse mode). sin θ = 0.5 is true at 30° and at 150°, and again every 360°. The calculator returns the principal value and writes out the full family.
Inverse trig conventions
Each inverse function returns one angle from a fixed range: sin⁻¹ gives −90° to 90°, cos⁻¹ gives 0° to 180°, tan⁻¹ gives strictly between −90° and 90°. For cot⁻¹ this calculator uses the 0°-to-180° range found in most textbooks, so cot⁻¹ of a negative number is an obtuse angle; some software returns the negative angle instead. sin⁻¹ and cos⁻¹ only accept values from −1 to 1, and csc⁻¹ and sec⁻¹ only accept values of at least 1 in size; anything else has no real angle, and the calculator says so.
A caution on radians
Radian input is a decimal, so π/6 typed as 0.5236 is very slightly off 30° and will show decimals rather than exact values. Type it to ten decimal places (0.5235987756) and the calculator recognises it as π/6.
Frequently asked questions
▸What is sin 30°, cos 30° and tan 30°?
sin 30° = 1/2, cos 30° = √3/2 ≈ 0.866025, and tan 30° = √3/3 ≈ 0.577350. They come from the 30-60-90 triangle with sides 1, √3 and 2.
▸Why is tan 90° undefined?
tan θ = sin θ / cos θ, and cos 90° = 0, so the division has no answer. As θ approaches 90° from below, tan θ grows without limit.
▸How do I find the reference angle?
Reduce the angle to between 0° and 360°, then measure to the nearest x-axis: θ in quadrant I, 180° − θ in II, θ − 180° in III, 360° − θ in IV.
▸Why does sin⁻¹(0.5) give only 30° when 150° works too?
An inverse function has to return one answer, so sin⁻¹ is restricted to −90° to 90°. Every solution is 30° + 360°k or 150° + 360°k, which the calculator lists.
▸How do I convert degrees to radians?
Multiply by π/180. So 45° = π/4 ≈ 0.785398 rad, and 1 radian ≈ 57.2958°.
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